We formulated and analyzed the problem of blind demixing of filtered diffused graph signals, an extension of blind demixing of time (or spatial) domain signals to graphs. Envisioned application domains encompass a particular class of bilinear inverse problems, where the observed graph signal is modeled as a linear combination of diffusion processes driven by a number of sources.
While the graph signal observations are bilinear functions of the filter coefficients and the sparse input signal, we rely on the frequency interpretation of graph signals and filters to show that they are also linearly related to the entries of a lifted rank-one, row-sparse matrix. Accordingly, the blind demixing problem of graph signals was tackled via rank and sparsity minimization subject to linear constraints, an inverse problem amenable to convex relaxations offering provable recovery guarantees under simplifying assumptions.
The theoretical results reveal key features driving the demixing performance including the number of sources and graph nodes, the sparsity levels, the filter orders, as well as spectral properties of the underlying graphs and their pairwise coherences. These spectral properties are captured via conveniently defined concentration parameters.
Numerical tests validated the theoretical claims and demonstrated that the proposed approach offers satisfactory recovery performance, even for settings well beyond the scope of the analysis.
Building on the insights gained here, it is of interest to develop spectral graph theoretical tools to facilitate identifying the key relationships between the (e.g., topological) properties of graphs and the resulting recovery performance. Future research includes also estimating the shift operator employing network topology inference [1]}, [2]}, [3]}.
